Superconductivity and Ginzburg-Landau theory

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15 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Superconductivity and Ginzburg-Landau theory
Stefan K¨olling
Universit
¨
at Bonn
Seminar on Quantum field theoretical methods for condensed
matter systems,July 07 2006
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Structure
1
Meissner-Effect London-Equation
2
Hubbard Stratonovich transformation
3
Application to BCS-Hamiltonian
4
Ginzburg-Landau theory
5
Anderson-Higgs effect
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect
Superconductivity is the consequence of an electron-phonon
interaction.
One of the most striking features of a superconductor is the
Meissner effect.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
For the equation of motion for an electron a filed E is:
m
d
dt
v = e
d
dt
E
d
dt
j =
￿￿￿￿
j=nev
ne
2
m
d
dt
E
Because of:
E =
−1
c
d
dt
A−￿Φ
We obtain the London-ansatz for j (Φ = 0):
London-ansatz
j = −
n
s
e
2
mc
A
with n
s
the density of superconducting electrons.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
For the equation of motion for an electron a filed E is:
m
d
dt
v = e
d
dt
E
d
dt
j =
￿￿￿￿
j=nev
ne
2
m
d
dt
E
Because of:
E =
−1
c
d
dt
A−￿Φ
We obtain the London-ansatz for j (Φ = 0):
London-ansatz
j = −
n
s
e
2
mc
A
with n
s
the density of superconducting electrons.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
Take the rotation:
∂xj = −
n
s
e
2
mc
∂xA = −
n
s
e
2
mc
B
Because of Maxwell equation ∂xB =

c
j and ∂B = 0:
∂x∂xB =

c
∂xj =
4πn
s
e
2
mc
2
B
= −ΔB+∂(∂B) = −ΔB
it follows:
ΔB =
4πn
s
e
2
mc
2
B Δj =
4πn
s
e
2
mc
2
j
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
Take the rotation:
∂xj = −
n
s
e
2
mc
∂xA = −
n
s
e
2
mc
B
Because of Maxwell equation ∂xB =

c
j and ∂B = 0:
∂x∂xB =

c
∂xj =
4πn
s
e
2
mc
2
B
= −ΔB+∂(∂B) = −ΔB
it follows:
ΔB =
4πn
s
e
2
mc
2
B Δj =
4πn
s
e
2
mc
2
j
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
Inspect this equation in the case of:
No superconductor at z<0
A superconductor at z>0

2
∂z
2
B(z) =
4πn
s
e
2
mc
2
B(z)
Solution
B(z) = B
o
e

z
λ
L
λ
L
=
￿
me
2
4πn
s
e
2
λ
L
:London penetration depth.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Meissner-Effect and London-Equation
Inspect this equation in the case of:
No superconductor at z<0
A superconductor at z>0

2
∂z
2
B(z) =
4πn
s
e
2
mc
2
B(z)
Solution
B(z) = B
o
e

z
λ
L
λ
L
=
￿
me
2
4πn
s
e
2
λ
L
:London penetration depth.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Hubbard Stratonovich transformation maps interacting fermion
systems to non-interacting fermions moving in an effective field.
→Interacting has to contain fermion bilinears
H = H
o
+H
I
with H
I
= −g
￿
d
3
xA
+
(x)A(x)
examples A(x) = Ψ

(x)Ψ

(x) or A(x) = S

(x)
Replacement:
−gaA
+
(x)A(x) →A
+
(x)Δ(x) +
¯
ΔA(x) +
¯
Δ(x)Δ(x)
g
Like in mean field theories.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Hubbard Stratonovich transformation maps interacting fermion
systems to non-interacting fermions moving in an effective field.
→Interacting has to contain fermion bilinears
H = H
o
+H
I
with H
I
= −g
￿
d
3
xA
+
(x)A(x)
examples A(x) = Ψ

(x)Ψ

(x) or A(x) = S

(x)
Replacement:
−gaA
+
(x)A(x) →A
+
(x)Δ(x) +
¯
ΔA(x) +
¯
Δ(x)Δ(x)
g
Like in mean field theories.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Δ = Δ
1
+i Δ
2
¯
Δ = Δ
1
−i Δ
2
￿

1

2
e


2
1

2
2
)
g
= πg

￿
dΔd
¯
Δ
2πig
e
¯
ΔΔ
g
= 1
Generalize to Δ(x,t):
￿
D[Δ,
¯
Δ] exp(−
￿
d
3
x
β
￿
0

¯
Δ(x,τ)Δ(x,τ)
g
) = 1
D[Δ,
¯
Δ] ≡
￿
τj
¯
dΔ(x
j
,τ)dΔ(x
j
,τ)
N
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Δ = Δ
1
+i Δ
2
¯
Δ = Δ
1
−i Δ
2
￿

1

2
e


2
1

2
2
)
g
= πg

￿
dΔd
¯
Δ
2πig
e
¯
ΔΔ
g
= 1
Generalize to Δ(x,t):
￿
D[Δ,
¯
Δ] exp(−
￿
d
3
x
β
￿
0

¯
Δ(x,τ)Δ(x,τ)
g
) = 1
D[Δ,
¯
Δ] ≡
￿
τj
¯
dΔ(x
j
,τ)dΔ(x
j
,τ)
N
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Z =
￿
D[c,¯c]e

β
￿
0
dτ[¯c(∂
τ
+h)c+H
I
]
h = ￿
a
δ
ab
By introducing a 1 we obtain:
Z =
￿
D[c,¯c]
￿
D[Δ,
¯
Δ]e

β
￿
0
dτ[¯c(∂
τ
+h)c+H
￿
I
]
H
￿
I
=
￿
d
3
x[
¯
ΔΔ
g
−g
¯
A(x)A(x)]
We now shift the Δ-field:
Δ(x) →Δ(x) +gA(x)
¯
Δ(x) →
¯
Δ(x) +g
¯
A(x)
⇒H
￿
I
=
￿
d
3
x[
¯
A(x)Δ(x) +
¯
ΔA(x) +
¯
Δ(x)Δ(x)
g
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
Z =
￿
D[c,¯c]e

β
￿
0
dτ[¯c(∂
τ
+h)c+H
I
]
h = ￿
a
δ
ab
By introducing a 1 we obtain:
Z =
￿
D[c,¯c]
￿
D[Δ,
¯
Δ]e

β
￿
0
dτ[¯c(∂
τ
+h)c+H
￿
I
]
H
￿
I
=
￿
d
3
x[
¯
ΔΔ
g
−g
¯
A(x)A(x)]
We now shift the Δ-field:
Δ(x) →Δ(x) +gA(x)
¯
Δ(x) →
¯
Δ(x) +g
¯
A(x)
⇒H
￿
I
=
￿
d
3
x[
¯
A(x)Δ(x) +
¯
ΔA(x) +
¯
Δ(x)Δ(x)
g
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
First result
We have absorbed the interaction,replacing it by an effective
action which couples to the fermion bilinear A.
Z =
￿
D[Δ,
¯
Δ]e

￿
d
3
xdτ
¯
ΔΔ
g
￿
D[c,¯c]e

￿
S
￿
S =
β
￿
0
dτ¯c∂
τ
c +H
eff
[Δ,
¯
Δ]
H
eff
[Δ,
¯
Δ] = H
o
+
￿
d
3
x[
¯
A(x)Δ(x) +
¯
ΔA(x)]
Please note that
￿
S is quadratic in the fermion operators.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
First result
We have absorbed the interaction,replacing it by an effective
action which couples to the fermion bilinear A.
Z =
￿
D[Δ,
¯
Δ]e

￿
d
3
xdτ
¯
ΔΔ
g
￿
D[c,¯c]e

￿
S
￿
S =
β
￿
0
dτ¯c∂
τ
c +H
eff
[Δ,
¯
Δ]
H
eff
[Δ,
¯
Δ] = H
o
+
￿
d
3
x[
¯
A(x)Δ(x) +
¯
ΔA(x)]
Please note that
￿
S is quadratic in the fermion operators.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
￿
D[c,¯c]e

￿
S
= det[∂
τ
+h
eff
[Δ,
¯
Δ]]
with h
eff
the matrix representation of H
eff
.
This leads to:
Z =
￿
D[Δ,
¯
Δ]e
−S
eff
[Δ,
¯
Δ]
S
eff
[Δ,
¯
Δ] =
￿
d
3
xdτ
¯
ΔΔ
g
−lndet[∂
τ
+h
eff
[Δ,
¯
Δ]]
=
￿
d
3
xdτ
¯
ΔΔ
g
−Tr ln[∂
τ
+h
eff
[Δ,
¯
Δ]]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Hubbard Stratonovich transformation
￿
D[c,¯c]e

￿
S
= det[∂
τ
+h
eff
[Δ,
¯
Δ]]
with h
eff
the matrix representation of H
eff
.
This leads to:
Z =
￿
D[Δ,
¯
Δ]e
−S
eff
[Δ,
¯
Δ]
S
eff
[Δ,
¯
Δ] =
￿
d
3
xdτ
¯
ΔΔ
g
−lndet[∂
τ
+h
eff
[Δ,
¯
Δ]]
=
￿
d
3
xdτ
¯
ΔΔ
g
−Tr ln[∂
τ
+h
eff
[Δ,
¯
Δ]]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
BCS-Hamiltonian
H =
￿

￿

c
+

c


g
o
V
A
+
A
A =
￿
k,|￿
k
|<ω
D
c
−k↓
c
k↑
A
+
=
￿
k,|￿
k
|<ω
D
c
+
−k↑
c
+
−k↓
By using the results of the previous section wse obtain:
Z =
￿
D[Δ,
¯
Δ,c,¯c]e
−S
S =
β
￿
0
dτ [
￿

¯c

(∂
τ
+￿
k
)c

+
¯
ΔA+
¯
AΔ+
¯
ΔΔ
g
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
BCS-Hamiltonian
H =
￿

￿

c
+

c


g
o
V
A
+
A
A =
￿
k,|￿
k
|<ω
D
c
−k↓
c
k↑
A
+
=
￿
k,|￿
k
|<ω
D
c
+
−k↑
c
+
−k↓
By using the results of the previous section wse obtain:
Z =
￿
D[Δ,
¯
Δ,c,¯c]e
−S
S =
β
￿
0
dτ [
￿

¯c

(∂
τ
+￿
k
)c

+
¯
ΔA+
¯
AΔ+
¯
ΔΔ
g
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
Introduce a Nambu notation:
S =
β
￿
0
dτ [
￿
k
¯
Ψ
k
(∂
τ
+h
k

k
+
¯
ΔΔ
g
]
Ψ
k
=
￿
c
k↑
¯c
−k↓
￿
h
k
=
￿
￿
k
Δ(τ)
¯
Δ(τ) −￿
k
￿
Again we can integrate out the fermionic contribution:
Z =
￿
D[Δ,
¯
Δ]e
−S
eff
[Δ,
¯
Δ]
S
eff
[Δ,
¯
Δ] =
β
￿
0

¯
ΔΔ
g
+
￿
k
Tr ln(∂
τ
+h
k
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
Introduce a Nambu notation:
S =
β
￿
0
dτ [
￿
k
¯
Ψ
k
(∂
τ
+h
k

k
+
¯
ΔΔ
g
]
Ψ
k
=
￿
c
k↑
¯c
−k↓
￿
h
k
=
￿
￿
k
Δ(τ)
¯
Δ(τ) −￿
k
￿
Again we can integrate out the fermionic contribution:
Z =
￿
D[Δ,
¯
Δ]e
−S
eff
[Δ,
¯
Δ]
S
eff
[Δ,
¯
Δ] =
β
￿
0

¯
ΔΔ
g
+
￿
k
Tr ln(∂
τ
+h
k
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
To proceed we must invoke some approximation,we expect that
fluctuations will be small
→integral will be dominated by minimal value of S
eff
→saddlepoint approximation Z = e
−S
eff

o
,
¯
Δ
o
]
Expect that Δ
o
is independent of τ because of translational
invariance.
Ψ
k
(τ) =
1

β
￿
n
Ψ
k
e
−i ω
n
τ
with the Matsubara frequencies ω
n
= (2n +1)
π
β
det[∂
τ
+h
k
] =
￿
n
det[−i ω
n
+h
k
] =
￿
n

2
n
+￿
2
k
+|Δ|
2
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
To proceed we must invoke some approximation,we expect that
fluctuations will be small
→integral will be dominated by minimal value of S
eff
→saddlepoint approximation Z = e
−S
eff

o
,
¯
Δ
o
]
Expect that Δ
o
is independent of τ because of translational
invariance.
Ψ
k
(τ) =
1

β
￿
n
Ψ
k
e
−i ω
n
τ
with the Matsubara frequencies ω
n
= (2n +1)
π
β
det[∂
τ
+h
k
] =
￿
n
det[−i ω
n
+h
k
] =
￿
n

2
n
+￿
2
k
+|Δ|
2
]
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
Inserting this into S
eff
yields:
S
eff
β
= −T
￿
kn
ln[ω
2
n
+￿
2
k
+|Δ|
2
] +
Δ
2
g
= F
eff
Minimizing wrt Δ:
∂F
eff
∂Δ
= −T
￿
kn
Δ
ω
2
n
+E
2
k
+
Δ
g
= 0
BCS Gap equation
1
g
=
1
β
￿
kn
1
ω
2
n
+E
2
k
E
k
=
￿
￿
2
k
+|Δ|
2
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
Inserting this into S
eff
yields:
S
eff
β
= −T
￿
kn
ln[ω
2
n
+￿
2
k
+|Δ|
2
] +
Δ
2
g
= F
eff
Minimizing wrt Δ:
∂F
eff
∂Δ
= −T
￿
kn
Δ
ω
2
n
+E
2
k
+
Δ
g
= 0
BCS Gap equation
1
g
=
1
β
￿
kn
1
ω
2
n
+E
2
k
E
k
=
￿
￿
2
k
+|Δ|
2
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
Inserting this into S
eff
yields:
S
eff
β
= −T
￿
kn
ln[ω
2
n
+￿
2
k
+|Δ|
2
] +
Δ
2
g
= F
eff
Minimizing wrt Δ:
∂F
eff
∂Δ
= −T
￿
kn
Δ
ω
2
n
+E
2
k
+
Δ
g
= 0
BCS Gap equation
1
g
=
1
β
￿
kn
1
ω
2
n
+E
2
k
E
k
=
￿
￿
2
k
+|Δ|
2
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Application to BCS-Hamiltonian
From this follows (after some work):
Equations for Δ and T
c
Δ = 2ω
D
e

1
gN(0)
T
c
=
e

Ψ(
1
2
)

ω
D
e

1
gN(0)
With N:density of states and Ψ(z) =
d
dz
lnΓ(z) the digamma
function.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Since transition is continuous close to T
c
,expand S
eff
for small Δ.
S
eff
[Δ,
¯
Δ] =
β
￿
0

¯
ΔΔ
g
+
￿
k
Tr ln(∂
τ
+h
k
)
with ∂
τ
+h
k
=
￿

τ
+￿
k
Δ(τ)
¯
Δ(τ) ∂
τ
−￿
k
￿
= G
−1
G
−1
:=
￿
[G
p
o
]
−1
Δ
¯
Δ [G
h
o
]
−1
￿
= G
−1
o
￿
1 +G
o
￿
0 Δ
¯
Δ 0
￿￿
Tr ln(G
−1
) = Tr ln(G
−1
0
) −
1
2
Tr[G
o
￿
0 Δ
¯
Δ 0
￿
]
2
+...
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Tr(G
p
o
ΔG
h
o
Δ) =
￿
kk
￿
G
p
o
< k|Δ|k
￿
> G
h
o
< k
￿
|
¯
Δ|k >
=
￿￿￿￿
q=k−k
￿
￿
q
Δ
q
¯
Δ
−q
1
βL
d
￿
k
G
p
o
(k)G
h
o
(k +q)
￿
￿￿
￿
Π(q)pairing susceptibility
Apart from the term Tr lnG
−1
o
S
eff
contains:
S
eff

￿
ω
n
q
[
1
g
+Π(ω
n
,q)]|Δ
ω
n
,q
|
2
+
/
O(Δ
4
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Tr(G
p
o
ΔG
h
o
Δ) =
￿
kk
￿
G
p
o
< k|Δ|k
￿
> G
h
o
< k
￿
|
¯
Δ|k >
=
￿￿￿￿
q=k−k
￿
￿
q
Δ
q
¯
Δ
−q
1
βL
d
￿
k
G
p
o
(k)G
h
o
(k +q)
￿
￿￿
￿
Π(q)pairing susceptibility
Apart from the term Tr lnG
−1
o
S
eff
contains:
S
eff

￿
ω
n
q
[
1
g
+Π(ω
n
,q)]|Δ
ω
n
,q
|
2
+
/
O(Δ
4
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Approximate Π
Π(ω
n
,q) = Π(0,0) +
q
2
2

2
q
Π(0,0) +
/
O(ω
n
,q
4
)
Transform to position representation (and include a term of order
Δ
4
):
S
eff
⊃ β
￿
d
d
r
￿
t
2
|Δ|
2
+
K
2
|∂Δ|
2
+u|Δ|
4
￿
t
2
=
1
g
+Π(0,0) K = ∂
q
Π(0,0) > 0 u > 0
We again assume that Z will be dominated by the minimal action:
∂Δ = 0
δS
eff
δ|Δ|
!
￿￿￿￿
= 0 =
δ
δ|Δ|
(
t
2
|Δ|
2
+u|Δ|
4
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Approximate Π
Π(ω
n
,q) = Π(0,0) +
q
2
2

2
q
Π(0,0) +
/
O(ω
n
,q
4
)
Transform to position representation (and include a term of order
Δ
4
):
S
eff
⊃ β
￿
d
d
r
￿
t
2
|Δ|
2
+
K
2
|∂Δ|
2
+u|Δ|
4
￿
t
2
=
1
g
+Π(0,0) K = ∂
q
Π(0,0) > 0 u > 0
We again assume that Z will be dominated by the minimal action:
∂Δ = 0
δS
eff
δ|Δ|
!
￿￿￿￿
= 0 =
δ
δ|Δ|
(
t
2
|Δ|
2
+u|Δ|
4
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
Approximate Π
Π(ω
n
,q) = Π(0,0) +
q
2
2

2
q
Π(0,0) +
/
O(ω
n
,q
4
)
Transform to position representation (and include a term of order
Δ
4
):
S
eff
⊃ β
￿
d
d
r
￿
t
2
|Δ|
2
+
K
2
|∂Δ|
2
+u|Δ|
4
￿
t
2
=
1
g
+Π(0,0) K = ∂
q
Π(0,0) > 0 u > 0
We again assume that Z will be dominated by the minimal action:
∂Δ = 0
δS
eff
δ|Δ|
!
￿￿￿￿
= 0 =
δ
δ|Δ|
(
t
2
|Δ|
2
+u|Δ|
4
)
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
⇒|Δ|(t +4u|Δ|
2
) = 0 |Δ| =
￿
0 t > 0
￿
t
4u
t < 0
For t < 0 U(1) symmetry is spontaneously broken.
Δ = |Δ|e
Φ
→Φ field remains massless/gapless.
Goldstone theorem
Every time a continous global symmetry gets spontaneous broken
there exists a gapless exitation →Goldstone mode.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
⇒|Δ|(t +4u|Δ|
2
) = 0 |Δ| =
￿
0 t > 0
￿
t
4u
t < 0
For t < 0 U(1) symmetry is spontaneously broken.
Δ = |Δ|e
Φ
→Φ field remains massless/gapless.
Goldstone theorem
Every time a continous global symmetry gets spontaneous broken
there exists a gapless exitation →Goldstone mode.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
By calculating Π(0,0) we can relate t with T:
T
c
= πω
D
exp(−
1
N(0)g
)
t
2
≈ N(0)
T −T
c
T
c
The Ginzburg-Landau theory of superconductors was known (1956)
before BCS-theory (1957) and predicted the right results.
Parameters are unknown if you start with a GL theory but
meaningfull predictions are possible.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Ginzburg-Landau theory
By calculating Π(0,0) we can relate t with T:
T
c
= πω
D
exp(−
1
N(0)g
)
t
2
≈ N(0)
T −T
c
T
c
The Ginzburg-Landau theory of superconductors was known (1956)
before BCS-theory (1957) and predicted the right results.
Parameters are unknown if you start with a GL theory but
meaningfull predictions are possible.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
Inclusion of em fields via minimal coupling:
∂ ￿→∂ +ieA
L
em
=
−1
4
F
µν
F
µν
F
µν
= ∂
µ
A
n
u −∂
ν
A
µ
→Z =
￿
DA
￿
D[Δ,
¯
Δ]e
−S
eff
Where S
eff
gets modified:
S
eff
= β
￿
dr
￿
t
2
|Δ|
2
+
K
2
|(∂ +i 2eA)Δ|
2
+u|Δ|
4
+
1
2
(∂xA)
2
￿
where we used A
o
= Φ = 0
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
All terms in this Lagrangian are gauge invariant under local gauge
transformations:
A ￿→A−∂Φ(r)
Δ ￿→e
−2ieΦ(r)
Δ
Write:
Δ(r) = |Δ(r)|e
−2ieΦ(r)
and choose a gauge (unitary gauge):
A ￿→A−∂Φ(r) Δ￿→|Δ|
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
All terms in this Lagrangian are gauge invariant under local gauge
transformations:
A ￿→A−∂Φ(r)
Δ ￿→e
−2ieΦ(r)
Δ
Write:
Δ(r) = |Δ(r)|e
−2ieΦ(r)
and choose a gauge (unitary gauge):
A ￿→A−∂Φ(r) Δ￿→|Δ|
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
this results in the action:
S
eff
= β
￿
dr
￿
t
2
|Δ|
2
+
K
2
(∂|Δ|)
2

4e
2
K|Δ|
2
2
A
2
+u|Δ|
4
+
1
2
(∂xA)
2
￿
below T
c
Δ ￿=0 ⇒m
2
A
= 4e
2
K|Δ|
2
￿== 0.
Anderson-Higgs effect
The goldstone mode gets eaten by the gauge field which aquires a
mass.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
this results in the action:
S
eff
= β
￿
dr
￿
t
2
|Δ|
2
+
K
2
(∂|Δ|)
2

4e
2
K|Δ|
2
2
A
2
+u|Δ|
4
+
1
2
(∂xA)
2
￿
below T
c
Δ ￿=0 ⇒m
2
A
= 4e
2
K|Δ|
2
￿== 0.
Anderson-Higgs effect
The goldstone mode gets eaten by the gauge field which aquires a
mass.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
The m
2
A
-term is responsible for London equation,because
minimisation wrt A:
∂x (∂xA)
￿
￿￿
￿
B
+m
2
A
A = 0
With the help of Maxwell equation ∂xB =

c
j we obtain:
j = −
cm
2
A

A
This was the basis for the London equation.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Anderson-Higgs effect
The m
2
A
-term is responsible for London equation,because
minimisation wrt A:
∂x (∂xA)
￿
￿￿
￿
B
+m
2
A
A = 0
With the help of Maxwell equation ∂xB =

c
j we obtain:
j = −
cm
2
A

A
This was the basis for the London equation.
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
Conclusions
We saw how a interacting fermionic system can be rewritten
as a system noninteracting fermions in a bosonic field
We could compute several important quantities like T
c
or the
gap Δ
A Ginzburg-Landau ansatz with in orderparameter < Ψ

Ψ

showed how symmetry breaking occurs
We saw an example of Goldstones theorem
By gauging the symmetry we obtained an example of
Anderson-Higgs mechanism which is also important in high
energy physics
Meissner-Effect London-Equation
Hubbard Stratonovich transformation
Application to BCS-Hamiltonian
Ginzburg-Landau theory
Anderson-Higgs effect
literature
P.Coleman 04,chapter 12
Czycholl,Theoretische Festk¨orperphysik